Teleportation-based quantum state tomography

TL;DR

The authors address the problem of fully reconstructing an arbitrary $n$-qubit density matrix via quantum teleportation. Their approach, termed teleportation-based quantum state tomography, uses Bell state measurements and a minimal set of four known single-qubit input states to generate a complete linear system that determines all density-matrix elements from Bob’s post-teleportation qubits. They demonstrate explicit reconstruction for two- and three-qubit cases and extend the framework to general $n$-qubit states, including distributed settings where multiple parties cooperate. The method provides a conceptually distinct QST tool that complements traditional techniques, with potential resource reductions and clear avenues for future work in higher-spin and continuous-variable regimes.

Abstract

We explicitly show that the quantum teleportation protocol can be employed to completely reconstruct arbitrary two- and three-qubit density matrices. We also extend the present analysis to n-qubit density matrices. The only quantum resources needed to implement the teleportation-based quantum state tomography protocol are the ability to make Bell measurements and the ability to prepare a few different single qubit states to be teleported from Alice to Bob.

Figures (3)

• Figure 1: (color online) The teleportation-based QST starts by Alice preparing an appropriate input state $\rho_A$. Subsequently, she realizes a Bell measurement (BM) on her two qubits ($A$ and $1$), informing Bob of the obtained result as well as the state describing initially qubit $A$. With this information, Bob knows that his qubit is $\varrho_2$ [Eq. (\ref{['stepE']})]. Repeating this process for a few different inputs, Bob can fully reconstruct the initial state describing qubits $1$ and $2$, i.e. $\rho_{12}$, with the knowledge of all states $\varrho_2$ associated with each different input state $\rho_A$.
• Figure 2: (color online) The three-qubit teleportation-based QST. Alice prepares two input states ($\rho_{A_1}$ and $\rho_{A_3}$) and then realizes independent BMs on each pair of qubits as shown in the figure, informing Bob of the obtained results and the states describing initially qubits $A_1$ and $A_3$. With this information, Bob knows that his qubit is given by $\varrho_2$ [Eq. (\ref{['stepE3']})]. Repeating this process for a few different combinations of pairs of input states $\rho_{A_1}$ and $\rho_{A_3}$, Bob can reconstruct the initial state describing $\rho_{123}$ if he knows all the states $\varrho_2$ associated with each different pair of input states.
• Figure 3: (color online) The $n$-qubit teleportation-based QST. Alice prepares $n-1$ input states ($\rho_{A_1}, \ldots, \rho_{A_{n-1}}$) and then realizes independent BMs on each pair of qubits as indicated in the figure, informing Bob of the results obtained and the states describing initially qubits $A_1, \ldots, A_{n-1}$. With this information, Bob knows that his qubit (denoted by $n$ in the figure) is given by the generalized version of $\varrho_2$ [Eq. (\ref{['stepE3']})]. Repeating this process for several different arrangements of $n-1$ inputs, Bob can reconstruct the initial state describing the $n$-qubit state if he knows all states $\varrho_2$ related to each different $n-1$ input state arrangement teleported by Alice.